Olympiad problems

2021 Kazakhstan National Olympiad, 4

Tags: geometry

Given acute triangle $ABC$ with circumcircle $\Gamma$ and altitudes $AD, BE, CF$, line $AD$ cuts $\Gamma$ again at $P$ and $PF, PE$ meet $\Gamma$ again at $R, Q$. Let $O_1, O_2$ be the circumcenters of $\triangle BFR$ and $\triangle CEQ$ respectively. Prove that $O_{1}O_{2}$ bisects $\overline{EF}$.

2023 MOAA, 7

Tags:

Written in mm/dd format, a date is called [i]cute[/i] if the month is divisible by the day. For example, the date [i]cute[/i] is a [i]cute[/i] date because $8$ is divisible by $2$. Find the number of [i]cute[/i] dates in a year. [i]Proposed by Andy Xu[/i]

2012 Bosnia and Herzegovina Junior BMO TST, 2

Tags: transformation , combinatorics , digit

Let $\overline{abcd}$ be $4$ digit number, such that we can do transformations on it. If some two neighboring digits are different than $0$, then we can decrease both digits by $1$ (we can transform $9870$ to $8770$ or $9760$). If some two neighboring digits are different than $9$, then we can increase both digits by $1$ (we can transform $9870$ to $9980$ or $9881$). Can we transform number $1220$ to: $a)$ $2012$ $b)$ $2021$

2016 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , geometry , 3d geometry , prism

Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

1973 Putnam, B3

Tags: number theory , prime number , polynomial

Consider an integer $p>1$ with the property that the polynomial $x^2 - x + p$ takes prime values for all integers $x$ such that $0\leq x <p$. Show that there is exactly one triple of integers $a, b, c$ satisfying the conditions: $$b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.$$

2004 Belarusian National Olympiad, 4

Tags: number theory

For a positive integer $A = \overline{a_n ...a_1a_0}$ with nonzero digits which are not all the same ($n \ge 0$), the numbers $A_k = \overline{a_{n-k}...a_1a_0a_n ...a_{n-k+1}}$ are obtained for $k = 1,2,...,n$ by cyclic permutations of its digits. Find all $A$ for which each of the $A_k$ is divisible by $A$.

2021 Canadian Mathematical Olympiad Qualification, 2

Tags: algebra , system of equations

Determine all integer solutions to the system of equations: \begin{align*} xy + yz + zx &= -4 \\ x^2 + y^2 + z^2 &= 24 \\ x^{3} + y^3 + z^3 + 3xyz &= 16 \end{align*}

2018 239 Open Mathematical Olympiad, 10-11.1

Tags: combinatorial geometry , geometry

Prove that in any convex polygon where all pairwise distances between vertices are distinct, there exists a vertex such that the closest vertex of the polygon is adjacent to it. [i]Proposed by D. Shiryayev, S. Berlov[/i]

2006 All-Russian Olympiad, 5

Tags: number theory

Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1<a_2<...<a_{10}$. For every $k$, denote by $b_k$ the greatest divisor of $a_k$ such that $b_k<a_k$. Assume that $b_1>b_2>...>b_{10}$. Show that $a_{10}>500$.

2015 AMC 10, 5

Tags:

David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place? $\textbf{(A) } \text{David} \qquad\textbf{(B) } \text{Hikmet} \qquad\textbf{(C) } \text{Jack} \qquad\textbf{(D) } \text{Rand} \qquad\textbf{(E) } \text{Todd} $

2024 Yasinsky Geometry Olympiad, 4

Tags: geometry , construction , incenter

Let $ \omega $ be the circumcircle of triangle $ ABC $, where $ AB > AC $. Let $ N $ be the midpoint of arc $ \smile\!BAC $, and $ D $ a point on the circle $ \omega $ such that $ ND \perp AB $. Let $ I $ be the incenter of triangle $ ABC $. Reconstruct triangle $ ABC $, given the marked points $ A, D, $ and $ I $. Proposed by Oleksii Karlyuchenko and Hryhorii Filippovskyi

2005 China Second Round Olympiad, 1

Tags: geometry , incenter , circumcircle

In $\triangle ABC$, $AB>AC$, $l$ is a tangent line of the circumscribed circle of $\triangle ABC$, passing through $A$. The circle, centered at $A$ with radius $AC$, intersects $AB$ at $D$, and line $l$ at $E, F$. Prove that lines $DE, DF$ pass through the incenter and an excenter of $\triangle ABC$ respectively.

1996 Turkey Junior National Olympiad, 2

Tags: geometry , geometric transformation

Write out the positive integers consisting of only $1$s, $6$s, and $9$s in ascending order as in: $1,6,9,11,16,\dots$. a. Find the order of $1996$ in the sequence. b. Find the $1996$th term in the sequence.

1997 Tournament Of Towns, (526) 3

Tags: area , geometry

The vertical diameter of a circle is moved a centimetres to the right, and the horizontal diameter of this circle is moved $b$ centimetres up. These two lines divide the circle into four pieces. Consider the sum of the areas of the largest and the smallest pieces, and the sum of the areas of the other two pieces. Find the difference between these two sums. (G Galperin, NB Vassiliev)

2006 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , limit , trigonometry

Compute $\displaystyle\lim_{x\to 0}\dfrac{e^{x\cos x}-1-x}{\sin(x^2)}.$

2008 Moldova Team Selection Test, 3

Tags: circumcircle , geometry

Let $ \omega$ be the circumcircle of $ ABC$ and let $ D$ be a fixed point on $ BC$, $ D\neq B$, $ D\neq C$. Let $ X$ be a variable point on $ (BC)$, $ X\neq D$. Let $ Y$ be the second intersection point of $ AX$ and $ \omega$. Prove that the circumcircle of $ XYD$ passes through a fixed point.

2018 India IMO Training Camp, 1

Tags: rectangle , combinatorics , tiling

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

2009 India Regional Mathematical Olympiad, 6

Tags: modular arithmetic , number theory

In a book with page numbers from $ 1$ to $ 100$ some pages are torn off. The sum of the numbers on the remaining pages is $ 4949$. How many pages are torn off?

2014 HMNT, 3

Tags: hmmt , greatest common divisor

Compute the greatest common divisor of $4^8 - 1$ and $8^{12} - 1$.

1994 Hong Kong TST, 1

Tags: inequalities

Suppose, $x, y, z \in \mathbb{R}_+$ such that $xy+yz+zx=1$. Prove that, \[x(1-y^2)(1-z^2)+y(1-z^2)(1-x^2)+z(1-x^2)(1-y^2)\leq \frac{4\sqrt{3}}{9}\]

2022 Yasinsky Geometry Olympiad, 3

Tags: geometry , isosceles right triangle , angle

In an isosceles right triangle $ABC$ with a right angle $C$, point $M$ is the midpoint of leg $AC$. At the perpendicular bisector of $AC$, point $D$ was chosen such that $\angle CDM = 30^o$, and $D$ and $B$ lie on different sides of $AC$. Find the angle $\angle ABD$. (Volodymyr Petruk)

2004 Poland - Second Round, 3

Tags: induction , number theory

Determine all sequences $ a_1,a_2,a_3,...$ of $ 1$ and $ \minus{}1$ such that $ a_{mn}\equal{}a_ma_n$ for all $ m,n$ and among any three successive terms $ a_n,a_{n\plus{}1},a_{n\plus{}2}$ both $ 1$ and $ \minus{}1$ occur.

2021 Irish Math Olympiad, 10

Tags: combinatorics , combinatorial geometry , centroid , geometry

Let $P_{1}, P_{2}, \ldots, P_{2021}$ be 2021 points in the quarter plane $\{(x, y): x \geq 0, y \geq 0\}$. The centroid of these 2021 points lies at the point $(1,1)$. Show that there are two distinct points $P_{i}, P_{j}$ such that the distance from $P_{i}$ to $P_{j}$ is no more than $\sqrt{2} / 20$.

2016 Philippine MO, 1

Tags: algebra , invariant , discriminant , quadratic

The operations below can be applied on any expression of the form $ax^2+bx+c$. $(\text{I})$ If $c \neq 0$, replace $a$ by $4a-\frac{3}{c}$ and $c$ by $\frac{c}{4}$. $(\text{II})$ If $a \neq 0$, replace $a$ by $-\frac{a}{2}$ and $c$ by $-2c+\frac{3}{a}$. $(\text{III}_t)$ Replace $x$ by $x-t$, where $t$ is an integer. (Different values of $t$ can be used.) Is it possible to transform $x^2-x-6$ into each of the following by applying some sequence of the above operations? $(\text{a})$ $5x^2+5x-1$ $(\text{b})$ $x^2+6x+2$

2010 Korea National Olympiad, 1

Tags: modular arithmetic , number theory , logarithm

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.